Question

Expressing a Function as a Composition Express the function in the form $f\circ g\circ h$.

$G\left(x\right)={\left(4+\sqrt[3]{x}\right)}^9$

Short answer

The required functions are $f\left(x\right)=x^9,g\left(x\right)=4+x,h\left(x\right)=\sqrt[3]{x}$.

Step-by-step solution

Step 1. Composition of functions.

Composition of functions is defined as:

$\left(f\circ g\circ h\right)\left(x\right)=f\left(g\left(h\left(x\right)\right)\right)$

Step 2. Given information.

The given function is:

$G\left(x\right)={\left(4+\sqrt[3]{x}\right)}^9$

Step 3. Express the function as composition of functions.

Since the formula for function$G\left(x\right)$ says to take cube root, then add in 4 and then take exponent 9.

Let,$h\left(x\right)=\sqrt[3]{x},g\left(x\right)=4+x$ and $f\left(x\right)=x^9$. Then,

$$\begin{gathered} \left(f\circ g\circ h\right)\left(x\right)=f\left(g\left(h\left(x\right)\right)\right) \\ =f\left(g\left(\sqrt[3]{x}\right)\right)\left[\because h\left(x\right)=\sqrt[3]{x}\right] \\ =f\left(4+\sqrt[3]{x}\right)\left[\because g\left(x\right)=4+x\right] \\ ={\left(4+\sqrt[3]{x}\right)}^9\left[\because f\left(x\right)=x^9\right] \\ =G\left(x\right) \end{gathered}$$

Thus, the required functions are $f\left(x\right)=x^9,g\left(x\right)=4+x,h\left(x\right)=\sqrt[3]{x}$.